Finding The Equation Of A Line Perpendicular To 3x + 5y = -9 And Passing Through (3, 0)
In mathematics, determining the equation of a line that is perpendicular to a given line and passes through a specific point is a fundamental concept in coordinate geometry. This task involves understanding the relationship between the slopes of perpendicular lines and utilizing point-slope form to construct the desired equation. Let's dive into a comprehensive exploration of this topic.
Understanding the Fundamentals of Perpendicular Lines
To effectively tackle the problem of finding the equation of a perpendicular line, it's crucial to grasp the fundamental concepts underlying perpendicularity. Perpendicular lines are lines that intersect at a right angle (90 degrees). A key property of perpendicular lines lies in their slopes. The slopes of two perpendicular lines are negative reciprocals of each other. This means that if one line has a slope of m, the slope of a line perpendicular to it will be -1/m. Understanding this relationship is essential for solving problems involving perpendicular lines.
Before we delve into the specifics of finding the equation, let's take a moment to recap the concept of slope. The slope of a line is a measure of its steepness and direction. It represents the change in the y-coordinate for every unit change in the x-coordinate. Mathematically, the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is given by the formula:
m = (y2 - y1) / (x2 - x1)
The slope plays a vital role in determining the equation of a line. When we know the slope of a line, we can use various forms, such as the slope-intercept form or the point-slope form, to write its equation. The slope-intercept form is particularly useful when the y-intercept (the point where the line crosses the y-axis) is known, while the point-slope form is beneficial when a specific point on the line is given.
In the context of perpendicular lines, the negative reciprocal relationship between their slopes is a cornerstone principle. If we know the slope of the given line, we can readily find the slope of any line perpendicular to it by simply taking the negative reciprocal. This understanding lays the foundation for finding the equation of a perpendicular line that satisfies specific conditions, such as passing through a given point. The ability to manipulate slopes and utilize them effectively is a crucial skill in coordinate geometry and is essential for tackling a wide range of problems involving lines and their relationships.
Determining the Slope of the Given Line
The first step in finding the equation of a line perpendicular to a given line is to determine the slope of the given line. In this case, the given line is represented by the equation 3x + 5y = -9. To find the slope, we need to rewrite the equation in slope-intercept form, which is y = mx + b, where m represents the slope and b represents the y-intercept. This form explicitly reveals the slope of the line, making it easy to identify.
To convert the given equation into slope-intercept form, we need to isolate y on one side of the equation. Let's walk through the steps:
- Subtract 3x from both sides of the equation: 5y = -3x - 9
- Divide both sides of the equation by 5: y = (-3/5)x - 9/5
Now, the equation is in slope-intercept form (y = mx + b). By comparing this equation with the general form, we can identify the slope of the given line. In this case, the slope (m) is -3/5. This value is crucial because it allows us to determine the slope of the line perpendicular to it.
Remember that the slopes of perpendicular lines are negative reciprocals of each other. Therefore, to find the slope of the perpendicular line, we need to take the negative reciprocal of -3/5. The negative reciprocal is obtained by flipping the fraction and changing its sign. In this case, the negative reciprocal of -3/5 is 5/3. This is the slope of the line we are trying to find.
Understanding how to manipulate equations to find the slope is a fundamental skill in algebra and coordinate geometry. The ability to convert equations into slope-intercept form is particularly useful, as it directly reveals the slope and y-intercept of the line. This information is essential for a variety of applications, including finding the equation of perpendicular lines, determining the parallelism of lines, and graphing linear equations. Once we have the slope of the given line, we can easily find the slope of the perpendicular line, paving the way for constructing its equation.
Finding the Slope of the Perpendicular Line
Having determined the slope of the given line, the next crucial step is to find the slope of the line perpendicular to it. As we discussed earlier, the slopes of perpendicular lines are negative reciprocals of each other. This means that to find the slope of the perpendicular line, we need to take the negative reciprocal of the slope of the given line.
The slope of the given line, which we found in the previous step, is -3/5. To find the negative reciprocal, we simply flip the fraction and change its sign. Flipping -3/5 gives us -5/3, and changing the sign makes it 5/3. Therefore, the slope of the line perpendicular to the given line is 5/3.
This concept of negative reciprocals is fundamental in coordinate geometry and plays a vital role in understanding the relationships between lines. Whenever two lines are perpendicular, their slopes will always be negative reciprocals of each other. This relationship provides a quick and easy way to determine the slope of a perpendicular line if the slope of the original line is known.
The ability to find the slope of a perpendicular line is essential for a variety of applications. It allows us to construct lines that intersect at right angles, which is crucial in fields such as engineering, architecture, and computer graphics. For example, when designing a building, architects need to ensure that walls are perpendicular to the floor. Similarly, in computer graphics, creating realistic images often involves drawing lines and surfaces that are perpendicular to each other.
In this specific problem, we need to find the equation of a line that is perpendicular to the given line and passes through a specific point. Now that we have determined the slope of the perpendicular line (5/3), we are well-equipped to use this information, along with the given point, to construct the equation of the line. The next step involves utilizing the point-slope form of a linear equation, which is particularly useful when we know the slope of a line and a point that it passes through. This will allow us to write the equation of the perpendicular line in a clear and concise manner.
Using the Point-Slope Form
Now that we have the slope of the perpendicular line (5/3) and a point it passes through ((3, 0)), we can use the point-slope form of a linear equation to find its equation. The point-slope form is a powerful tool that allows us to write the equation of a line when we know its slope and a point on the line. It is expressed as:
y - y1 = m(x - x1)
where:
- m is the slope of the line
- (x1, y1) is a point on the line
In our case, we have m = 5/3 and (x1, y1) = (3, 0). Substituting these values into the point-slope form, we get:
y - 0 = (5/3)(x - 3)
This equation represents the line that is perpendicular to the given line and passes through the point (3, 0). However, to present the equation in a more standard form, such as slope-intercept form or standard form, we can simplify it further.
To simplify the equation, we can distribute the 5/3 on the right side:
y = (5/3)x - (5/3)(3)
y = (5/3)x - 5
Now, the equation is in slope-intercept form (y = mx + b), where we can clearly see the slope (5/3) and the y-intercept (-5). This form is useful for graphing the line and understanding its behavior.
Alternatively, we can convert the equation into standard form, which is Ax + By = C, where A, B, and C are integers. To do this, we can multiply both sides of the equation by 3 to eliminate the fraction:
3y = 5x - 15
Then, rearrange the terms to get the standard form:
5x - 3y = 15
This is the equation of the line in standard form. It is important to note that both the slope-intercept form and the standard form represent the same line, just in different formats. The choice of which form to use often depends on the specific application or the desired level of clarity.
The point-slope form is a versatile tool that is widely used in coordinate geometry. It provides a direct way to write the equation of a line when we know its slope and a point on the line. This is particularly useful in situations where we are given a point and a condition that determines the slope, such as perpendicularity or parallelism. By using the point-slope form, we can efficiently find the equation of the line and then manipulate it into other forms, such as slope-intercept form or standard form, as needed.
Identifying the Correct Equation
After simplifying the equation obtained from the point-slope form, we have arrived at two possible forms: slope-intercept form (y = (5/3)x - 5) and standard form (5x - 3y = 15). To identify the correct equation from the given options, we need to compare our derived equations with the provided choices.
The given options are:
- 3x + 5y = -9
- 3x + 5y = 9
- 5x - 3y = -15
- 5x - 3y = 15
By comparing our derived equation in standard form (5x - 3y = 15) with the options, we can see that it matches the fourth option exactly. Therefore, the correct equation of the line that is perpendicular to the given line (3x + 5y = -9) and passes through the point (3, 0) is:
5x - 3y = 15
It is important to note that the other options do not satisfy the conditions of the problem. The first two options, 3x + 5y = -9 and 3x + 5y = 9, represent lines that have the same slope as the given line, and therefore are parallel to it, not perpendicular. The third option, 5x - 3y = -15, represents a line that is perpendicular to the given line, but it does not pass through the point (3, 0). To verify this, we can substitute x = 3 and y = 0 into the equation: 5(3) - 3(0) = 15, which is not equal to -15.
This process of comparing derived results with given options is a crucial step in problem-solving, especially in mathematics. It ensures that the final answer satisfies all the conditions of the problem and that no errors were made during the solution process. By carefully examining the options and comparing them with our derived equation, we can confidently identify the correct answer.
In this case, the correct equation, 5x - 3y = 15, represents a line that is both perpendicular to the given line and passes through the specified point. This equation satisfies all the requirements of the problem and is therefore the final solution.
Conclusion
In conclusion, finding the equation of a line that is perpendicular to a given line and passes through a specific point involves a series of steps rooted in the principles of coordinate geometry. We began by understanding the fundamental relationship between the slopes of perpendicular lines, recognizing that they are negative reciprocals of each other. This understanding allowed us to determine the slope of the perpendicular line by taking the negative reciprocal of the slope of the given line.
Next, we utilized the point-slope form of a linear equation, which is a powerful tool for constructing the equation of a line when we know its slope and a point it passes through. By substituting the slope of the perpendicular line and the coordinates of the given point into the point-slope form, we obtained an equation that represents the desired line.
We then simplified the equation, transforming it into both slope-intercept form and standard form. These different forms of the equation provide different perspectives on the line's properties, such as its slope, y-intercept, and relationship to the coordinate axes. Finally, we compared our derived equations with the given options to identify the correct equation that satisfies all the conditions of the problem.
The ability to find the equation of a perpendicular line is a valuable skill in mathematics and has applications in various fields, including geometry, calculus, and physics. It is a fundamental concept that builds upon the understanding of linear equations, slopes, and the relationships between lines. By mastering this skill, students can tackle a wide range of problems involving lines, angles, and geometric figures.
In this specific problem, the correct equation of the line that is perpendicular to the given line (3x + 5y = -9) and passes through the point (3, 0) is 5x - 3y = 15. This equation represents a line that intersects the given line at a right angle and contains the point (3, 0), fulfilling all the requirements of the problem.
The process of solving this problem demonstrates the power of combining geometric concepts with algebraic techniques. By understanding the relationships between lines and their equations, we can effectively solve complex problems and gain a deeper appreciation for the interconnectedness of mathematics.
